Peijian Shi

• Ph.D.
• Professor (Associate) at Nanjing Forestry University, P.R. China

Stomatal pore area and density determine the capacity for gas exchange between the leaf interior and the atmosphere. Stomatal area is given by the profile formed by two guard cells, and the cumulative stomatal area characterizes the area of leaf surface occupied by stomata. The areas of all stomata captured in a micrograph are sorted in ascending o...

Key message The Montgomery equation, which assumes a proportional relationship between the tepal area and the product of the tepal length and width, is validated using data drawn from four Magnolia species. Abstract An important metric of floral non-reproductive size is individual petal or tepal area (A). The Montgomery equation (ME) estimates A b...

This new package predicts the occurrence dates (in day-of-year) of spring phenological events. Three methods, including the accumulated degree days (ADD) method, the accumulated days transferred to a standardized temperature (ADTS) method, and the accumulated developmental progress (ADP) method, were included. See Shi et al. (2017a, b).

The concept of a geometric series (GS) plays an important role in mathematics. However, it has been neglected in describing biological size series. Herein, we show that a GS describes the nonreproductive (perianth) parts of the flowers of four Magnoliaceae species and two Rosaceae species and the leaves of 60 Alangium chinense and 60 Shibataea chin...

Total leaf area per shoot (AT) can reflect the photosynthetic capacity of a shoot. A prior study hypothesized that AT is proportional to the product of the sum of the individual leaf widths per shoot (LKS) and the maximum individual leaf length per shoot (WKS), referred to as the Montgomery–Koyama–Smith equation (MKSE). However, empirical evidence...

The size of floral organs is closely related to the successful reproduction of plants, and corolla size is, to some extent, indicative of the size of floral organs. Petals are considered homologous to leaves, so we also attempted to estimate the area of a single petal using a method (i.e., the Montgomery equation) that is typically employed for est...

Premise Although previous studies have reported a positive correlation between leaf dry mass per unit area (LMA) and mean leaf thickness (LT), the LMA versus LT scaling relationship has not been determined due to limited sample sizes, despite its importance in estimating leaf bulk tissue density (mass per unit volume). Methods This issue was addre...

Skewness, a measure of the asymmetry of a distribution, is frequently employed to reflect a biologically important property. Another statistic, the Gini coefficient (GC), originally used to measure economic inequality, has been validated in measuring the inequality of biological size distributions. Given that the GC and skewness control overlapping...

The Montgomery equation (ME) assumes that leaf area (A) is proportional to the product of leaf length (L) and width (W). Leaf shape is found to determine the ME’s proportionality coefficient, i.e., the Montgomery parameter (MP). However, prior work seldom reported the influence of tree size (reflected by the diameter at breast height, DBH) on leaf...

The performance of various life processes in arthropods is greatly influenced by temperature. The survival of arthropods is temperature-dependent, which must be accounted for when predicting and simulating outbreaks of pest population under global warming scenarios. However, mathematical models for describing temperature-dependent arthropod surviva...

Photosynthetic light response curves serve as powerful mathematical tools for quantitatively describing the rate of photosynthesis of plants in response to changes in irradiance. However, in practical applications, the daunting task of selecting an appropriate nonlinear model to accurately fit these curves persists as a significant challenge. Thus,...

Cotyledons play an important role in early seedling establishment. However, relative to primary leaves, cotyledons tend to have a different investment-on-return strategy. To detect the potential differences in the mass (M) versus area (A) scaling relationships between cotyledons and primary leaves in different light environments, a total of 75 Acer...

Background and aims The Montgomery-Koyama-Smith (MKS) equation predicts that total leaf area per shoot is proportional to the product of the sum of individual leaf widths and maximum individual leaf length, which has been validated for some herbaceous and woody plants. The equation is also predicted to be valid in describing the relationship betwee...

The "leafing intensity premium" hypothesis proposes that leaf size results from natural selection acting on different leafing intensities, i.e., the number of leaves per unit shoot volume or mass. The scaling relationships among various above-ground functional traits in the context of this hypothesis are important for understanding plant growth and...

There is a lack of research on whether tree size affects lamina and petiole biomass allocation patterns, whereas the trade‐off between leaf biomass allocated to the lamina and the petiole is of significance when considering the hydraulic and mechanical function of the leaf as a whole. Here, Camptotheca acuminata Decne was selected for study because...

Total leaf area per plant is an important measure of the photosynthetic capacity of an individual plant that together with plant density drives the canopy leaf area index, that is, the total leaf area per unit ground area. Because the total number of leaves per plant (or per shoot) varies among conspecifics and among mixed species communities, this...

Previous studies have validated a performance equation (PE) and its generalized version (GPE) in describing the rotated and right-shifted Lorenz curves of organ size (e.g., leaf area and fruit volume) distributions of herbaceous plants. Nevertheless, there are still two questions that have not been adequately addressed by prior work: (i) whether th...

A 2-dimensional (2D) egg-shape equation can be used to construct a 3D egg geometry based on the hypothesis that an egg is a solid of revolution, which helps to calculate egg volume and surface area. The parameters in the 2D egg-shape equation are potentially valuable for providing a clue to the ecology and evolution of avian eggs. In this study, th...

Background and aims: Leaf area (A) is a crucial indicator of the photosynthetic capacity of plants. The Montgomery equation (ME), which hypothesizes that A is proportional to the product of leaf length (L) and width (W), is a valid tool for nondestructively measuring A for many broad-leaved plants. At present, the methods used to compute L and W f...

In nature, the two-dimensional (2D) profiles of fruits from many plants often resemble ellipses. However, it remains unclear whether these profiles strictly adhere to the ellipse equation, as many natural shapes resembling ellipses are actually better described as superellipses. The superellipse equation, which includes an additional parameter n co...

The Lorenz curve is used to describe the relationship between the cumulative proportion of household income and the number of households of an economy. The extent to which the Lorenz curve deviates from the line of equality (i.e., y = x) is quantified by the Gini coefficient. Prior models are based on the simulated and empirical data of income dist...

The vascular veins in photosynthetic leaves play an important role in transporting water and sugars throughout the plant body, and their venation pattern and vein density determine the hydraulic efficiency of the leaf. Likewise, stomatal density (SD) can influence photosynthetic gas exchange. However, the correlation between leaf vein density and S...

The scaling relationship between functional traits of leaf sheath and leaf lamina in bamboo is of significant importance in regulating bamboo growth, yet related studies are lacking. In this study, we collected a total of 2006 leaves from six bamboo species (including Pleioblastus argenteostriatus, Pleioblastus chino var. hisauchii, Pleioblastus fo...

The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the ori...

Improved quantitative methods for forest management, yields and carbon fixation are needed for bamboo. We report here a novel scientific method for quantifying natural forms. Remarkably, the origin of this breakthrough lies in the observation that square bamboos (species of the genus Chimonobambusa) are superelliptic. Superellipses can be considere...

Synopsis The proportions in the size of the avian egg albumen, yolk, and shell are crucial for understanding bird survival and reproductive success because their relationships with volume and surface area can affect ecological and life history strategies. Prior studies have focused on the relationship between the albumen and the yolk, but little is...

Key message A significant scaling relationship between the perianth mass and area was detected using 306 flowers of Magnolia × soulangeana, and increases in mass failed to achieve proportional increases in area. Abstract The scaling relationship between leaf lamina mass and area has been explored extensively. However, this relationship for floral...

The number and composition of species in a community can be quantified with α-diversity indices, including species richness ( R ), Simpson’s index ( D ), and the Shannon–Wiener index ( H΄ ). In forest communities, there are large variations in tree size among species and individuals of the same species, which result in differences in ecological pro...

Foliage leaves are essential for plant survival and growth, and how plants allocate biomass to their leaves reveals their economic and ecological strategies. Prior studies have shown that leaf-age significantly influences leaf biomass allocation patterns. However, unravelling the effects of ontogeny on partitioning biomass remains a challenge becau...

In nature, the fruit shapes of many plants resemble avian eggs, a form extensively studied as solids of revolution. Despite this, the hypothesis that egg-shaped fruits are themselves solids of revolution remains unvalidated. To address this, 751 Cucumis melo L. var. agrestis Naud. fruits were photographed, and the two-dimensional (2D) boundary coor...

Stomata are involved in transpiration and CO2 uptake by mediating gas exchange between internal plant tissues and the atmosphere. The capacity for gas exchange depends on stomatal density (SD), stomatal size, and pore dimensions. Most published work on stomatal quantification has assumed that stomatal distribution and stomatal density are spatially...

Exploring the morphometric scaling relationships between lamina and petiole functional traits is important to our understanding of foliage and plant ecosystem dynamics. For this purpose, we examined the leaves of two evergreen species, i.e. Photinia × fraseri "Red Robin", which has relatively small leaves, and Photinia serratifolia, which has relat...

Measuring the inequality of leaf area distribution per plant (ILAD) can provide a useful tool for quantifying the influences of intra- and interspecific competition, foraging behavior of herbivores, and environmental stress on plants’ above-ground architectural structures and survival strategies. Despite its importance, there has been limited resea...

Key message A generalized performance equation is proposed to fit the Lorenz curve of the leaf size distribution of an individual plant and is validated using 12 individual bamboo plants. Abstract The goal of this study is to provide a rigorous tool to quantify the inequality of the leaf size distribution of an individual plant, thereby serving as...

α-diversity describes species diversity at local scales. The Simpson’s and Shannon–Wiener indices are widely used to characterize α-diversity based on species abundances within a fixed study site (e.g., a quadrat or plot). Although such indices provide overall diversity estimates that can be analyzed, their values are not spatially continuous nor a...

Although many fruit geometries resemble a solid of revolution, this assumption has rarely been rigorously examined. To test this assumption, 574 fruits of Canarium album (Lour.) DC. which appear to have an ellipsoidal shape, were examined to determine the validity of a general avian-based egg-shape equation, referred to as the explicit Preston equa...

Key message Stomatal area is proportional to the product of stomatal length and width with a proportionality coefficient exceeding π/4 (indicating a deviation from an elliptical shape) for four Magnoliaceae species. Abstract Stomatal size is an important factor affecting the photosynthetic and transpiration rates of vascular plants. To examine sto...

Lamina dry mass (LDM) per unit area is an important plant functional trait. However, it is time-consuming to dry leaves in practice. Previous studies have demonstrated that lamina fresh mass (LFM) is approximately proportional to LDM for some broad-leaved plants. However, those studies largely overlooked the influence of leaf age on the proportiona...

Egg geometry can be described using Preston's equation, which has seldom been used to calculate egg volume (V) and surface area (S) to explore S versus V scaling relationships. Herein, we provide an explicit re‐expression of Preston's equation (designated as EPE) to calculate V and S, assuming that an egg is a solid of revolution. The side (longitu...

People who take a walk in urban parks, including or adjacent to a water body such as a river, a pond, or a lake, usually suffer from mosquito bites in summer and early autumn. The insects can negatively affect the health and mood of these visitors. Prior studies about the effects of landscape composition on the abundance of mosquitos have usually t...

Preston's equation is a general model describing the egg shape of birds. The parameters of Preston's equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg's actual length to two. This method assumes that the straight line through the two points on an egg's profile separated by the maximum distance (i.e...

Stomatal density (SD) is important to photosynthetic rates. However, it is time-consuming to measure SD. Here, we provide a method for estimating SD based on the scaling relationship between SD and mean nearest neighbour distance (MNND) of sampled stomatal centres. 397 leaves from eight Magnoliaceae species were used for this study. For each leaf,...

Plants have diverse leaf shapes that have evolved to adapt to the environments they have experienced over their evolutionary history. Leaf shape and leaf size can greatly influence the growth rate, competitive ability, and productivity of plants. However, researchers have long struggled to decide how to properly quantify the complexity of leaf shap...

The Lamé curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lamé curve in the polar coordinate system by introducing additional parameters (n1, n2, n3; m): rφ=1Acosm4φn2+1Bsinm4φn3−1/n1, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar a...

“Diminishing returns” in leaf economics occurs when increases in lamina mass (M), which can either be represented by lamina dry mass (DM) or fresh mass (FM), fail to produce proportional increases in leaf surface area (A), such that the scaling exponent (α) for the M vs. A scaling relationship exceeds unity (i.e., α > 1.0). Prior studies have shown...

A uniform description of natural shapes and phenomena is an important goal in science. Such description should check some basic principles, related to 1) the complexity of the model, 2) how well its fits real objects, phenomena and data, and 3) a direct connection with optimization principles and the calculus of variations. In this article, we pres...

Leaf shape and size can vary between hybrids and their parents. However, this has seldom been quantitatively tested. Photinia × fraseri is an important landscaping plant in East Asia as a hybrid between evergreen shrubs P. glabra and P. serratifolia. Its leaf shape looks like that of P. serratifolia. To investigate leaf shape, we used a general equ...

Many geometries of plant organs can be described by the Gielis equation, a polar coordinate equation extended from the superellipse equation, . Here, r is the polar radius corresponding to the polar angle φ; m is a positive integer that determines the number of angles of the Gielis curve when φ ∈ [0 to 2π); and the rest of the symbols are parameter...

Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called ‘biogeom’ that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the pro...

The Montgomery equation predicts leaf area as the product of leaf length and width multiplied by a correction factor. It has been demonstrated to apply to a variety of leaf shapes. However, it is unknown whether tree size (measured as the diameter at breast height) affects leaf shape and size, or whether such variations in leaf shape can invalidate...

Recently, a universal equation by Narushin, Romanov, and Griffin (hereafter, the NRGE) was proposed to describe the shape of avian eggs. While NRGE can simulate the shape of spherical, ellipsoidal, ovoidal, and pyriform eggs, its predictions were not tested against actual data. Here, we tested the validity of the NRGE by fitting actual data of egg...

Sexual dimorphism in body size has been observed for many insect species. However, whether dimorphism influences the flight performance for closely related insects or between the genders of conspecifics has seldom been examined. Wing loading (i.e., body mass/total wing area, WL = M/A) is an important indicator of flight efficiency and maneuverabili...

Leaf lamina mass and area are closely correlated with the photosynthetic capacity and competitive ability of plants, whereas leaf age has been demonstrated to affect physiological processes such as photosynthesis. However, it remains unknown whether the lamina mass vs. area scaling relationship is influenced by leaf age, which is important for unde...

Most insects engage in winged flight. Wing loading, that is, the ratio of body mass to total wing area, has been demonstrated to reflect flight maneuverability. High maneuverability is an important survival trait, allowing insects to escape natural enemies and to compete for mates. In some ecological field experiments, there is a need to calculate...

Premise: Across species, main leaf vein density scales inversely with leaf area (A). Yet, minor vein density manifests no clear relationship with respect to A, despite having the potential to provide important insights into the trade-off among the investments in leaf mechanical support, hydraulics, and light interception. Methods: To examine thi...

Premise: The phenomenon called "diminishing returns" refers to the scaling relationship between lamina mass (M) vs. lamina area (A) in many species, i.e., M ∝ Aα>1 , where αis the scaling exponent exceeding unity. Prior studies have focused on the scaling relationships between lamina dry mass (DM) and A, or between fresh mass (FM) and A. However,...

Leaf shape is an important leaf trait, with ovate leaves common in many floras. Recently, a new leaf shape model (referred to as the MLRF equation) derived from temperature-dependent bacterial growth was proposed and demonstrated to be valid in describing leaf boundaries of many species with ovate leaf shape. The MLRF model’s parameters can provide...

Abstract: Many natural radial symmetrical shapes (e.g., sea stars) follow the Gielis equation (GE) or its twin equation (TGE). A supertriangle (three triangles arranged around a central polygon) represents such a shape, but no study has tested whether natural shapes can be represented as/are supertriangles or whether the GE or TGE can describe thei...

The shape of leaf laminae exhibits considerable diversity and complexity that reflects adaptations to environmental factors such as ambient light and precipitation as well as phyletic legacy. Many leaves appear to be elliptical which may represent a ‘default’ developmental condition. However, whether their geometry truly conforms to the ellipse equ...

Premise: Leaf mass (M) and lamina surface area (A) are important functional traits reported to obey a scaling relationship called "diminishing returns" (i.e., M ∝ Aα>1 ). Previous studies have focused primarily on eudicots and ignored whether the age of leaves affects the numerical value of the scaling exponent (i.e., α). Methods: The effect of...

Background and Aims In hierarchically reticulate venation patterns, smaller orders of veins form areoles in which stomata are located. This study aimed to quantify the spatial relationship among stomata at the areole level. Methods For each of 12 leaves of Michelia cavaleriei var. platypetala, we assumed that stomatal characteristics were symmetri...

Background and Aims Leaf size has considerable ecological relevance, making it desirable to obtain leaf size estimations for as many species worldwide as possible. Current global databases, such as TRY, contain leaf size data for approximately 30,000 species, which is only ca. 8% of known species worldwide. Yet, taxonomic descriptions exist for the...

Leaf shape is closely related to economics of leaf support and leaf functions, including light interception, water use, and CO2 uptake, so correct quantification of leaf shape is helpful for studies of leaf structure/function relationships. There are some extant indices for quantifying leaf shape, including the leaf width/length ratio (W/L), leaf s...

Many cross-sectional shapes of plants have been found to approximate a superellipse rather than an ellipse. Square bamboos, belonging to the genus Chimonobambusa (Poaceae), are a group of plants with round-edged square-like culm cross sections. The initial application of superellipses to model these culm cross sections has focused on Chimonobambusa...

Key message Using more than 10,000 bamboo leaves, we found that the scaling between leaf surface area and length follows a uniform power law relationship mainly relying on the degree of variation in leaf shape (reflected by the ratio of leaf width to length). Abstract A recent study based on leaf data of different plant taxa showed that the scalin...

Premise: The nondestructive measurement of leaf area is important for expediting data acquisition in the field. The Montgomery equation (ME) assumes that leaf area (A) is a proportional function of the product of leaf length (L) and width (W), i.e., A = cLW, where c is called the Montgomery parameter. The ME has been successfully applied to calcula...

The scaling relationship between either leaf dry or fresh mass (M) and surface area (A) can reflect the photosynthetic potential and efficiency of light harvesting in different broad-leaved plants. In growing leaves, lamina area expansion is typically finished before the completion of leaf biomass accumulation, thereby affecting the M vs. A scaling...

Pleioblastus pygmaeus (Miq.) Nakai, as a dwarf bamboo, providing a high ornamental value in landscaping. Furthermore its well-developed and intricate root system lends its use for soil consolidation and slope protection. A pot experiment was performed to study the impact of different light intensity on the growth characteristics of P. pygmaeus seed...

Leaves, as the most important photosynthetic organ of plants, are intimately associated with plant function and adaptation to environmental changes. The scaling relationship of the leaf dry mass (or the fresh mass) vs. leaf surface area has been referred to as “diminishing returns”, suggesting that the leaf area fails to increase in proportion to l...

Moso bamboo is widespread in natural forests and is cultivated over large areas in China. This study investigated how climate controls its distribution, about which little is known. We collected moso bamboo presence-absence data from 674 sites with long-term climate data in Mainland China. Generalized additive models that included location and clim...

Leaf bilateral asymmetry, an important indicator of leaves that tend to be affected by the above-ground architectural structure of plants and their environments (especially light), has been poorly studied. Taylor’s power law (TPL) describes a power-law relationship between the mean and variance of a non-negative random variable, and its exponent ha...

Leaf area is an important indicator of photosynthetic capacity in plants. Knowledge of the variation of leaf shape at an individual level or a population level can help by accurately calculate leaf area. Such calculations can help us to understand the life history strategies of plants, and allow us to better understand the influence of climate chan...

Key message A highly significant and positive scaling relationship between bamboo leaf dry mass and leaf surface area was observed; leaf shape (here, represented by the quotient of leaf width and length) had a significant influence on the scaling exponent of leaf dry mass vs. area. ContextThe scaling of leaf dry mass vs. leaf area is important for...

The leaves of vines exhibit a high degree of variability in shape, from simple oval to highly dissected palmatifid leaves. However, little is known about the extent of leaf bilateral symmetry in vines, how leaf perimeter scales with leaf surface area, and how this relationship depends on leaf shape. We studied 15 species of vines and calculated (i)...

Allelopathy plays an important role in plant-plant interactions, particularly through compounds released from litter exudates and decomposition. We conducted a greenhouse experiment to examine how additions of Cinnamomum septentrionale Hand. Mazz leaf litter (A) versus leaf litter aqueous extracts (B) can impact Eucalyptus grandis Hill ex Maid sapl...

Shapes, sizes and biomass investment per unit area (LMA) of vine leaves are characterized by high diversity that results in variation in leaf arrangement, light harvesting efficiency and photosynthetic activity. There exists a scaling relationship between leaf dry mass and surface area for many broad-leaved plants, and most estimates of the scaling...

There is convincing evidence for a scaling relationship between leaf dry weight (DW) and leaf surface area (A) for broad-leaved plants, and most estimates of the scaling exponent of DW vs. A are greater than unity. However, the scaling relationship of leaf fresh weight (FW) vs. A has been largely neglected. In the present study, we examined whether...

Key message Leaf area is proportional to the product of leaf length and width both for herbaceous and woody plants. The coefficient of proportionality falls into the range (1/2, π/4) for the investigated species. Abstract According to Thompson’s principle of similarity, the area of an object should be proportional to its length squared. However, l...

Taylor's power law (TPL) describes the scaling relationship between the temporal or spatial variance and mean of population densities by a simple power law. TPL has been widely testified across space and time in biomedical sciences, botany, ecology, economics, epidemiology, and other fields. In this paper, TPL is analytically reconfirmed by testify...

The mean and variance of ecological measures usually follow a power-law relationship, referred to as Taylor's power law (TPL). Leaves are important organs for photosynthesis, and leaf size is closely related to photosynthetic potential. Leaf size has different physical measures, such as leaf length, area, and fresh or dry weight. However, it has no...